标题： 凝聚态量子系统的能量色散曲率 显示英文标题

标题： Curvature of Energy Dispersions in Condensed Matter Quantum Systems

摘要： 石墨烯是一种具有可调电子特性的二维材料，在凝聚态物理和材料科学中具有重要意义。 在这项研究中，我们通过分析石墨烯在不同 onsite 势能值 M 下的高斯曲率和平均曲率，研究了石墨烯基态能量色散带的曲率。我们在周期性边界条件下使用石墨烯的哈密顿量进行分析。 我们对哈密顿矩阵进行了对角化，以在动量空间布里渊区的离散网格上获得特征值，从而构建基态能量色散带。 然后计算曲面微分，利用第一基本形式和第二基本形式的定义，在网格上的每个点计算高斯曲率和平均曲率。 这些值被整理成曲率图，并且我们绘制了每个图中最大高斯曲率和平均曲率的对数值与对应 M 的对数值。经过对这些图的分析，我们观察到直到 onsite 势能等于或大于石墨烯哈密顿量的最近邻跳跃参数时，高斯曲率和平均曲率大致保持不变，之后开始下降。 因此，本研究为进一步研究石墨烯哈密顿量提供了见解，并表明研究能量色散的平均曲率和高斯曲率可能是研究量子系统的一种可行方法。 进一步的研究方向可能包括其他哈密顿量的拓扑研究。 显示英文摘要

摘要： Graphene, a two-dimensional material with tunable electronic properties, holds significant importance in condensed matter physics and material science. In this study, we analyze the curvature of graphene's ground-state energy dispersion band by examining its Gaussian and mean curvature under varying on-site potential values, M, using graphene's Hamiltonian under periodic boundary conditions. We diagonalized the Hamiltonian matrix to obtain eigenvalues on a discretized grid for the Brillouin zone in momentum space, constructing the ground-state energy dispersion band. Surface differentials are then calculated to subsequently calculate Gaussian and mean curvature at each point in the grid using the definitions of the first and second fundamental forms. These values were collated to form curvature plots, and we plotted the logarithm of maximum Gaussian and mean curvature values from each plot against the corresponding values of the logarithm of M. After analyzing these plots, we observe that both the Gaussian and mean curvature remain approximately the same until the on-site potential is equal to or greater than the nearest-neighbor hopping parameter of the graphene Hamiltonian, at which point they begin to decline. Therefore, this study offers insights into studying the graphene Hamiltonian further and shows that studying the mean and Gaussian curvature of energy dispersions can prove to be a viable method of studying quantum systems. Further directions may include topological studies of other Hamiltonians.